Mathematics

19 pages, 4870 KiB

Open AccessArticle

On Non-Tensor Product Bivariate Fractal Interpolation Surfaces on Rectangular Grids

by Vasileios Drakopoulos and Polychronis Manousopoulos

Mathematics 2020, 8(4), 525; https://doi.org/10.3390/math8040525 - 03 Apr 2020

Cited by 7 | Viewed by 2350

Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal [...] Read more.

Some years ago, several authors tried to construct fractal surfaces which pass through a given set of data points. They used bivariable functions on rectangular grids, but the resulting surfaces failed to be continuous. A method based on their work for generating fractal interpolation surfaces is presented. Necessary conditions for the attractor of an iterated function system to be the graph of a continuous bivariable function which interpolates a given set of data are also presented here. Moreover, a comparative study for four of the most important constructions and attempts on rectangular grids is considered which points out some of their limitations and restrictions. Full article

(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)

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13 pages, 268 KiB

Open AccessArticle

The Mean Minkowski Content of Homogeneous Random Fractals

by Martina Zähle

Mathematics 2020, 8(6), 883; https://doi.org/10.3390/math8060883 - 01 Jun 2020

Cited by 2 | Viewed by 1530

Abstract

Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski [...] Read more.

Homogeneous random fractals form a probabilistic generalisation of self-similar sets with more dependencies than in random recursive constructions. Under the Uniform Strong Open Set Condition we show that the mean D-dimensional (average) Minkowski content is positive and finite, where the mean Minkowski dimension D is, in general, greater than its almost sure variant. Moreover, an integral representation extending that from the special deterministic case is derived. Full article

(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)

14 pages, 422 KiB

Open AccessFeature PaperArticle

Non-Stationary Fractal Interpolation

by Peter Massopust

Mathematics 2019, 7(8), 666; https://doi.org/10.3390/math7080666 - 25 Jul 2019

Cited by 13 | Viewed by 2510

Abstract

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps

{F_{k}}_{k \in N}

where each

F_{k}

maps

H (X) \to H (X)

and arises [...] Read more.

We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps

{F_{k}}_{k \in N}

where each

F_{k}

maps

H (X) \to H (X)

and arises from an iterated function system. Employing the recently-developed theory of non-stationary versions of fixed points and the concept of forward and backward trajectories, we present new classes of fractal functions exhibiting different local and global behavior and extend fractal interpolation to this new, more flexible setting. Full article

(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)

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18 pages, 363 KiB

Open AccessArticle

Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

by Mourad Ben Slimane, Moez Ben Abid, Ines Ben Omrane and Mohamad Maamoun Turkawi

Mathematics 2020, 8(7), 1179; https://doi.org/10.3390/math8071179 - 17 Jul 2020

Cited by 4 | Viewed by 1335

Abstract

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube

I^{2}

in

R^{2}

. Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) [...] Read more.

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube

I^{2}

in

R^{2}

. Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total

I^{2}

(or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal. Full article

(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)

16 pages, 331 KiB

Open AccessArticle

Periodic Intermediate β-Expansions of Pisot Numbers

by Blaine Quackenbush, Tony Samuel and Matt West

Mathematics 2020, 8(6), 903; https://doi.org/10.3390/math8060903 - 03 Jun 2020

Cited by 1 | Viewed by 2059

Abstract

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are

β

-shifts, namely transformations of the form [...] Read more.

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are

β

-shifts, namely transformations of the form

T_{β, α} : x \mapsto β x + α mod 1

acting on

[- α / (β - 1), (1 - α) / (β - 1)]

, where

(β, α) \in Δ

is fixed and where

Δ ≔ {(β, α) \in R^{2} : β \in (1, 2) and 0 \leq α \leq 2 - β}

. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of

(β, α)

such that

T_{β, α}

has the subshift of finite type property is dense in the parameter space

Δ

. Here, they proposed the following question. Given a fixed

β \in (1, 2)

which is the n-th root of a Perron number, does there exists a dense set of

α

in the fiber

{β} \times (0, 2 - β)

, so that

T_{β, α}

has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when

α = 0

to the case when

α \in (0, 2 - β)

. That is, we examine the structure of the set of eventually periodic points of

T_{β, α}

when

β

is a Pisot number and when

β

is the n-th root of a Pisot number. Full article

(This article belongs to the Special Issue Fractals: Geometry, Analysis and Mathematical Physics)

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